# Deepak DharIndian Institute of Science Education and Research, Pune | IISER · Department of Physics

Deepak Dhar

Ph.D. (Physics)

## About

210

Publications

55,755

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6,917

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Introduction

I work in the general area of statistical physics. My current interests are phase transitions in hard core models, percolation theory, application of statistical physics techniques to interdisciplinary problems.

Additional affiliations

November 2016 - present

August 1978 - October 2016

Education

September 1972 - May 1978

August 1970 - June 1972

August 1968 - June 1970

## Publications

Publications (210)

We study a system of equal-size circular disks, each with an asymmetrically placed pivot at a fixed distance from the center. The pivots are fixed at the vertices of a regular triangular lattice. The disks can rotate freely about the pivots, with the constraint that no disks can overlap with each other. Our Monte Carlo simulations show that the one...

In this article, I discuss the motion of point masses in nonrelativistic mechanics, when the interaction between them is purely the Newtonian gravitational interaction, with ≥ 2. The dynamical equations of motion cannot be solved in closed form, for general initial conditions, for any > 2. However, the qualitative behavior of the solutions can be u...

This article contains our comments and views on the status of the current understanding of phase transitions in systems in thermal equilibrium with only hard-core interactions, based on our work in this area. The equation of state for the hard sphere gas in d-dimensions is discussed. The universal repulsive Lee-Yang singularities in the complex act...

This article contains our comments and views on the status of the current understanding of phase transitions in systems with only hard-core interactions, based on our work in this area. The equation of state for the hard sphere gas in $d$-dimensions is discussed. The universal repulsive Lee-Yang singularities in the complex activity plane, and its...

We obtain the phase diagram of fully packed hard plates on the cubic lattice. Each plate covers an elementary plaquette of the cubic lattice and occupies its four vertices, with each vertex of the cubic lattice occupied by exactly one such plate. We consider the general case with fugacities sμ for “μ plates,” whose normal is the μ direction (μ=x,y,...

We study the phase diagram of a lattice gas of 2×2×1 hard plates on the three-dimensional cubic lattice. Each plate covers an elementary plaquette of the cubic lattice, with the constraint that a site can belong to utmost one plate. We focus on the isotropic system, with equal fugacities for the three orientations of plates. We show, using grand ca...

We study a system of equal-sized circular discs each with an asymmetrically placed pivot at a fixed distance from the center. The pivots are fixed at the vertices of a regular triangular lattice. The discs can rotate freely about the pivots, with the constraint that no discs can overlap with each other. Our Monte Carlo simulations show that the one...

Cellular automaton models provide simple minimal models to describe the salient features of many complex physical phenomena. In this article, I illustrate this with some examples: the sandpile model, Eulerian walkers model, and a model of fragmentation of ice sheet at termini of calving glaciers.KeywordsSandpile modelCellular automataPhysical syste...

We present a simple one-dimensional stochastic model with three control parameters and a surprisingly rich zoo of phase transitions. At each (discrete) site x and time t, an integer n(x,t) satisfies a linear interface equation with added random noise. Depending on the control parameters, this noise may or may not satisfy the detailed balance condit...

We present a simple 1-d stochastic model with two control parameters and a surprisingly rich zoo of phase transitions. At each (discrete) site $x$ and time $t$, there is an integer $n(x,t)$ that satisfies a linear interface equation with added random noise. Depending on the control parameters, this noise may or may not satisfy the detailed balance...

In a system of interacting thin rigid rods of equal length 2ℓ on a two-dimensional grid of lattice spacing a, we show that there are multiple phase transitions as the coupling strength κ=ℓ/a and the temperature are varied. There are essentially two classes of transitions. One corresponds to the Ising-type spontaneous symmetry-breaking transition an...

We study the entropy of a set of identical hard objects, of general shape, with each object pivoted on the vertices of a d -dimensional regular lattice of lattice spacing a , but can have arbitrary orientations. When the pivoting point is situated asymmetrically on the object, we show that there is a range of lattice spacings a , where in any orien...

A system of hard rigid rods of length k on hypercubic lattices is known to undergo two phase transitions when chemical potential is increased: from a low density isotropic phase to an intermediate density nematic phase, and on further increase to a high-density phase with no orientational order. In this paper, we argue that, for large k, the second...

We discuss the finite-size scaling of the ferromagnetic Ising model on random regular graphs. These graphs are locally tree-like, and in the limit of large graphs, the Bethe approximation gives the exact free energy per site. In the thermodynamic limit, the Ising model on these graphs show a phase transition. This transition is rounded off for fini...

In a system of interacting thin rigid rods of equal length $2 \ell$ on a two-dimensional grid of lattice spacing $a$, we show that there are multiple phase transitions as the coupling strength $\kappa=\ell/a$ and the temperature are varied. There are essentially two classes of transitions. One corresponds to the Ising-type spontaneous symmetry brea...

We study the entropy of a set of identical hard objects, of general shape, with each object pivoted on the vertices of a d-dimensional regular lattice of lattice spacing a, but can have arbitrary orientations. When the pivoting point is situated asymmetrically on the object, we show that there is a range of lattice spacings a, where in any orientat...

We discuss the finite-size scaling of the ferromagnetic Ising model on random regular graphs. These graphs are locally tree-like, and in the limit of large graphs, the Bethe approximation gives the exact free energy per site. In the thermodynamic limit, the Ising model on these graphs show a phase transition. This transition is rounded off for fini...

A system of hard rigid rods of length $k$ on hypercubic lattices is known to undergo two phases transitions when chemical potential is increased: from a low density isotropic phase to an intermediate density nematic phase, and on further increase to a high-density phase with no orientational order. In this paper, we argue that, for large $k$, the s...

We study the phase diagram of fully packed hard plates on a cubic lattice. Each plate covers the four corner vertices of a plaquette, and each vertex of the lattice is covered by exactly one plaquette, We consider the general case with fugacities $s_\mu$ for plates whose normal is the $\mu$ direction ($\mu = x,y,z$). At and close to the isotropic p...

We study the phase diagram of a system of $2\times 2\times 1$ hard plates on the three dimensional cubic lattice, {\em i.e.} a lattice gas of plates that each cover a single face of the cubic lattice and touch the four points of the corresponding square plaquette. We focus on the isotropic system, with equal fugacity for the three orientations of p...

We study the distribution of lengths and other statistical properties of worms constructed by Monte Carlo worm algorithms in the power-law three-sublattice ordered phase of frustrated triangular and kagome lattice Ising antiferromagnets. Viewing each step of the worm construction as a position increment (step) of a random walker, we demonstrate tha...

Chase-escape percolation is a variation of the standard epidemic spread models. In this model, each site can be in one of three states: unoccupied, occupied by a single prey, or occupied by a single predator. Prey particles spread to neighboring empty sites at rate p, and predator particles spread only to neighboring sites occupied by prey particle...

We determine the asymptotic behavior of the entropy of full coverings of a L×M square lattice by rods of size k×1 and 1×k, in the limit of large k. We show that full coverage is possible only if at least one of L and M is a multiple of k, and that all allowed configurations can be reached from a standard configuration of all rods being parallel, us...

We study relaxation of long-wavelength density perturbations in a one-dimensional conserved Manna sandpile. Far from criticality where correlation length ξ is finite, relaxation of density profiles having wave numbers k→0 is diffusive, with relaxation time τR∼k−2/D with D being the density-dependent bulk-diffusion coefficient. Near criticality with...

We determine the asymptotic behavior of the entropy of configurations of straight rigid rods of size $k\times 1$ and $1\times k$ that fully cover a finite $L \times M$ rectangular portion of the square lattice. We show that full coverage is possible only if at least one of $L$ and $M$ is a multiple of $k$, and that all allowed configurations can be...

We study relaxation of long-wavelength density perturbations in one dimensional conserved Manna sandpile. Far from criticality where correlation length $\xi$ is finite, relaxation of density profiles having wave numbers $k \rightarrow 0$ is diffusive, with relaxation time $\tau_R \sim k^{-2}/D$ with $D$ being the density-dependent bulk-diffusion co...

Chase-escape percolation is a variation of the standard epidemic spread models. In this model, each site can be in one of three states: unoccupied, occupied by a single prey, or occupied by a single predator. Prey particles spread to neighboring empty sites at rate $p$, and predator particles spread only to neighboring sites occupied by prey partic...

We consider the asymptotic shape of clusters in the Eden model on a d-dimensional hypercubical lattice. We discuss two improvements for the well-known upper bound to the growth velocity in different directions by that of the independent branching process (IBP). In the IBP, each cell gives rise to a daughter cell at a neighboring site at a constant...

Advantage begets further advantage - this effect, often called the Matthew effect, is observed in various forms in day-to-day life, most commonly in social and economic aspects. But can such an effect be seen in systems where the evolution is completely memory-less? In this work, we study the TASEP Speed Process, in which the dynamics of a special...

We discuss the approximate phenomenological description of the motion of a single second-class particle in a two-species totally asymmetric simple exclusion process (TASEP) on a 1D lattice. Initially, the second class particle is located at the origin and to its left, all sites are occupied with first class particles while to its right, all sites a...

We consider the asymptotic shape of clusters in the Eden model on a d-dimensional hypercubical lattice. We discuss two improvements for the well-known upper bound to the growth velocity in different directions by that of the independent branching process (IBP). In the IBP, each cell gives rise to a daughter cell at a neighboring site at a constant...

To obtain proper insight into how structure develops during a protein folding reaction, it is necessary to understand the nature and mechanism of the polypeptide chain collapse reaction which marks the initiation of folding. Here, the time-resolved fluorescence resonance energy transfer technique, in which the decay of the fluorescence light intens...

We study the phase diagram of a system of 2×2×2 hard cubes on a three-dimensional cubic lattice. Using Monte Carlo simulations, we show that the system exhibits four different phases as the density of cubes is increased: disordered, layered, sublattice ordered, and columnar ordered. In the layered phase, the system spontaneously breaks up into para...

We discuss the approximate phenomenological description of the motion of a single second-class particle in a two-species totally asymmetric simple exclusion process (TASEP) on a 1D lattice. Initially, the second class particle is located at the origin and to its left, all sites are occupied with first class particles while to its right, all sites a...

We study the phase diagram of a system of $2\times2\times2$ hard cubes on a three dimensional cubic lattice. Using Monte Carlo simulations, we show that the system exhibits four different phases as the density of cubes is increased: disordered, layered, sublattice ordered, and columnar ordered. In the layered phase, the system spontaneously breaks...

We study the distribution of lengths and other statistical properties of worms constructed by worm algorithms used in Monte Carlo simulations of frustrated triangular and kagome lattice Ising antiferromagnets, focusing on the behaviour of the associated persistence exponent $\theta$ in the critical phase associated with the two-step melting of thre...

There is a misconception, widely shared among physicists, that the equilibrium free energy of a one-dimensional classical model with strictly finite-ranged interactions, and at nonzero temperatures, cannot show any singularities as a function of the coupling constants. In this Letter, we discuss an instructive counterexample. We consider thin rigid...

We consider a minimalist model of overtaking dynamics in one dimension. On each site of a one-dimensional infinite lattice sits an agent carrying a random number specifying the agent's preferred velocity, which is drawn initially for each agent independently from a common distribution. The time evolution is Markovian, where a pair of agents at adja...

There is a misconception, widely shared amongst physicists, that the equilibrium free energy of a one-dimensional classical model with strictly finite-ranged interactions, and at non-zero temperatures, can not show any singularities as a function of the coupling constants. In this Letter, we discuss an instructive counter-example. We consider thin...

There is a misconception, widely shared amongst physicists, that the equilibrium free energy of a one-dimensional classical model with strictly finite-ranged interactions, and at non-zero temperatures , can not show any singularities as a function of the coupling constants. In this Letter, we discuss an instructive counterexample. We consider thin...

We consider a one-dimensional infinite lattice where at each site there sits an agent carrying a velocity, which is drawn initially for each agent independently from a common distribution. This system evolves as a Markov process where a pair of agents at adjacent sites exchange their positions with a specified rate, while retaining their respective...

We discuss the strategy that rational agents can use to maximize their expected long-term payoff in the co-action minority game. We argue that the agents will try to get into a cyclic state, where each of the $(2N +1)$ agent wins exactly $N$ times in any continuous stretch of $(2N+1)$ days. We propose and analyse a strategy for reaching such a cycl...

We discuss the strategy that rational agents can use to maximize their expected long-term payoff in the co-action minority game. We argue that the agents will try to get into a cyclic state, where each of the (2N + 1) agent wins exactly N times in any continuous stretch of (2N + 1) days. We propose and analyse a strategy for reaching such a cyclic...

In this article, I discuss the relationship of mathematics to the physical world, and to other spheres of human knowledge. In particular, I argue that Mathematics is created by human beings, and the number $\pi$ can not be said to have existed $100,000$ years ago, using the conventional meaning of the word `exist'.

We study the different phases of a system of monodispersed hard rods of length $k$ on a cubic lattice using an efficient cluster algorithm which can simulate densities close to the fully-packed limit. For $k\leq 4$, the system is disordered at all densities. For $k=5,6$, we find a single density-driven transition from a disordered phase to high den...

This viewpoint relates to the article by Wilkinson and Willemsen (1983 J Phys. A: Math. Gen.
16 3365), and was published as a part of a series of viewpoints celebrating 50 of the most
influential papers published in the Journal of Physics series, which is celebrating its 50th
anniversary.

These lectures provide an introduction to the directed percolation and directed animals problems, from a physicist's point of view. The probabilistic cellular automaton formulation of directed percolation is introduced. The planar duality of the diode-resistor-insulator percolation problem in two dimensions, and relation of the directed percolation...

We consider a toy model of interacting extrovert and introvert agents introduced earlier by Liu et al. (Europhys. Lett. 100 (2012) 66007). The number of extroverts, and introverts is N each. At each time step, we select an agent at random, and allow her to modify her state. If an extrovert is selected, she adds a link at random to an unconnected in...

We consider the response of a memoryless nonlinear device that converts an input signal $\xi(t)$ into an output $\eta(t)$ that only depends on the value of the input at the same time, $t$. For input Gaussian noise with power spectrum $1/f^{\alpha}$, the nonlinearity modifies the spectral index of the output to give a spectrum that varies as $1/f^{\...

We present simulations of the 1-dimensional Oslo rice pile model in which the critical height at each site is randomly reset after each toppling. We use the fact that the stationary state of this sandpile model is hyperuniform to reach system of sizes $> 10^7$. Most previous simulations were seriously flawed by important finite size corrections. We...

We consider the quantum-mechanical non-relativistic hydrogen atom. We show that for bound states with size much larger than the Bohr radius, one can construct a wave packet that is localized in space corresponding to a classical particle moving in a circular orbit.

We consider a toy model of interacting extrovert and introvert agents introduced earlier by Liu et al [Europhys. Lett. {\bf 100} (2012) 66007]. The number of extroverts, and introverts is $N$ each. At each time step, we select an agent at random, and allow her to modify her state. If an extrovert is selected, she adds a link at random to an unconne...

A system of $2\times d$ hard rectangles on square lattice is known to show
four different phases for $d \geq 14$. As the covered area fraction $\rho$ is
increased from $0$ to $1$, the system goes from low-density disordered phase,
to orientationally-ordered nematic phase, to a columnar phase with
orientational order and also broken translational in...

We study the directed Abelian sandpile model on a square lattice, with $K$
downward neighbors per site, $K > 2$. The $K=3$ case is solved exactly, which
extends the earlier known solution for the $K=2$ case. For $K>2$, the avalanche
clusters can have holes and side-branches and are thus qualitatively different
from the $K=2$ case where avalance clu...

Since their inception about a decade ago, dynamic networks which adapt to the
state of the nodes have attracted much attention. One simple case of such an
adaptive dynamics is a model of social networks in which individuals are
typically comfortable with a certain number of contacts, i.e., preferred
degrees. This paper is partly a review of earlier...

We consider a model of fragmentation of sheet by cracks that move with a
velocity in preferred direction, but undergo random transverse displacements as
they move. There is a non-zero probability of crack-splitting, and the split
cracks move independently. If two cracks meet, they merge, and move as a single
crack. In the steady state, there is non...

We show that critical exponents of the transition to columnar order in a {\em
mixture} of $2 \times 1$ dimers and $2 \times 2$ hard-squares on the square
lattice {\em depends on the composition of the mixture} in exactly the manner
predicted by the theory of Ashkin-Teller criticality, including in the
hard-square limit. This result settles the ques...

We propose an analytically tractable variation of the minority game in which
rational agents use probabilistic strategies. In our model, $N$ agents choose
between two alternatives repeatedly, and those who are in the minority get a
pay-off 1, others zero. The agents optimize the expectation value of their
discounted future pay-off, the discount par...

We study the growing patterns in the rotor-router model formed by adding $N$
walkers at the center of a $L \times L$ two-dimensional square lattice,
starting with a periodic background of arrows, and relaxing to a stable
configuration. The pattern is made of large number of triangular regions, where
in each region all arrows point in the same direc...

Discovering the mechanism underlying the ubiquity of $"1/f^{\alpha}"$ noise
has been a long--standing problem. The wide range of systems in which the
fluctuations show the implied long--time correlations suggests the existence of
some simple and general mechanism that is independent of the details of any
specific system. We argue here that a {\it m...

An interesting feature of growth in animals is that different parts of the
body grow at approximately the same rate. This property is called proportionate
growth. In this paper, we review our recent work on patterns formed by adding
$N$ grains at a single site in the abelian sandpile model. These simple models
show very intricate patterns, show pro...

We construct a class of assisted-hopping models in one dimension in which a particle can move only if it has exactly one occupied neighbour, or if it lies in an otherwise empty interval of length <= n + 1. We determine the exact steady state by a mapping to a gas of defects with only on-site interaction. We show that this system undergoes a phase t...

We construct a class of assisted hopping models in one dimension in which a
particle can move only if it does not lie in an otherwise empty interval of
length greater than $n+1$. We determine the exact steady state by a mapping to
a gas of defects with only on-site interaction. We show that this system
undergoes a phase transition as a function of...

This is a light-hearted take at the the second law of thermodynamics.

We study the phase transition from a nematic phase to a high-density
disordered phase in systems of long rigid rods of length $k$ on the square and
triangular lattices. We use an efficient Monte Carlo scheme that partly
overcomes the problem of very large relaxation times of nearly jammed
configurations. The existence of a continuous transition is...

This section of Resonance presents thought-provoking questions, and discusses answers a few months later. Readers are invited to send new questions, solutions to old ones and comments, to ‘Think It Over’, Resonance, Indian Academy of Sciences, Bangalore 560 080. Items illustrating ideas and concepts will generally be chosen.

We study a system of particles with nearest- and next-nearest-neighbor exclusion on the square lattice (hard squares). This system undergoes a transition from a fluid phase at low density to a columnar-ordered phase at high density. We develop a systematic high-activity perturbation expansion for the free energy per site about a state with perfect...

An important question in biology is how the relative size of different organs
is kept nearly constant during growth of an animal. This property, called
proportionate growth, has received increased attention in recent years. We
discuss our recent work on a simple model where this feature comes out quite
naturally from local rules, without fine tunin...

We relate properties of nearest-neighbour resonating valence bond (nnRVB)
wavefunctions for $SU(g)$ spin systems on two dimensional bipartite lattices to
those of fully-packed classical dimer models with potential energy $V$ on the
same lattice. We define a cluster expansion of $V$ in terms of $n$-body
potentials $V_n$, which are recursively determ...

We consider a directed abelian sandpile on a strip of size $2\times n$,
driven by adding a grain randomly at the left boundary after every $T$
time-steps. We establish the exact equivalence of the problem of mass
fluctuations in the steady state and the number of zeroes in the ternary-base
representation of the position of a random walker on a ring...

We present a Monte Carlo algorithm for studying the equilibrium
properties of hard rod fluids having only excluded volume interactions.
The algorithm does not suffer from slow-down due to jamming even at
densities close to the maximum possible. Implementing this algorithm on
a two dimensional square lattice, we show the existence of a transition
fr...

We study the patterns formed by adding N sand grains at a single site on an initial periodic background in the Abelian sandpile models, and relaxing the configuration. When the heights at all sites in the initial background are low enough, one gets patterns showing proportionate growth, with the diameter of the pattern formed growing as N(1/d) for...

We consider the spherical model on a spider-web graph. This graph is
effectively infinite-dimensional, similar to the Bethe lattice, but has loops.
We show that these lead to non-trivial corrections to the simple mean-field
behavior. We first determine all normal modes of the coupled springs problem on
this graph, using its large symmetry group. In...

We study a system of long rigid rods of fixed length k with only excluded volume interaction. We show that, contrary to the general expectation, the self-consistent field equations of the Bethe approximation do not give the exact solution of the problem on the Bethe lattice in this case. We construct a new lattice, called the random locally treelik...

The Silences of the Archives, the Reknown of the Story.
The Martin Guerre affair has been told many times since Jean de Coras and Guillaume Lesueur published their stories in 1561. It is in many ways a perfect intrigue with uncanny resemblance, persuasive deception and a surprizing end when the two Martin stood face to face, memory to memory, befor...

We study the Bethe approximation for a system of long rigid rods of fixed
length k, with only excluded volume interaction. For large enough k, this
system undergoes an isotropic-nematic phase transition as a function of density
of the rods. The Bethe lattice, which is conventionally used to derive the
self-consistent equations in the Bethe approxim...